Henrik JohanssonIPhT Saclay

June 27, 2011

Strings 2011 Uppsala

Based on work in collaboration with:Z.Bern, J.J.Carrasco, L.Dixon, R.Roiban

Lie Algebra Structures in Yang-Millsand Gravity Amplitudes

Strings June 27 2011 H. Johansson 2

Simple double-copy structure of gravityDuality between color and kinematics

Evidence at tree levelExplicit loop amplitudes with manifest duality

Amplitude UV behavior from dualityKinematic Lie algebra and Lagrangian formulation

Conclusion

Outline

Strings June 27 2011 H. Johansson 3

Einstein Gravity Feynman rules

After symmetrization ~ 100 terms !

=

=

de Donder gauge:

~103 terms

higher ordervertices…

Strings June 27 2011 H. Johansson 4

On-shell simplificationsGraviton plane wave:

=

Gravity scattering amplitude:

Yang-Mills polarization

Yang-Mills vertex

Yang-Mills amplitude

On-shell 3-graviton vertex:

On-shell gravity objects are “squares” of Yang-Mills objects !•holds for the entire S-matrix Bern, Carrasco, HJ [BCJ]

Strings June 27 2011 H. Johansson 5

Kawai-Lewellen-Tye Relations

gravity states areproducts of gaugetheory states:

Field theory limit ⇒ gravity theory ~ (gauge theory) × (gauge theory)

String theorytree-level identity: closed string ∼ (left open string) × (right open string)

|1〉grav = |1〉gauge ⊗ |1〉gauge

KLT relations emerge after nontrivial world-sheet integral identities

Strings June 27 2011 H. Johansson 6

Gravity should be cubic

Einstein gravity → cubic schematically:

Yang-Mills → cubic schematically:

schematic derivative

And gravity should be a double copy of a YM theory:

[BCJ]

Strings June 27 2011 H. Johansson 7

Gauge theory is the keyThe simplicity of gravity stems from a novel structure in Yang-Mills

•represent amplitudes using cubic graphs only:

Diagram numeratorssatisfy the algebra:

These are the same relations the color factors satisfy (Lie Algebra)Duality: color ↔ kinematics

Jacobiidentity

antisymmetry

propagators

color factors

numerators[BCJ]

Strings June 27 2011 H. Johansson 8

Gravity is a double copy

• The two numerators can belong to different theories:

• Gravity amplitudes are obtained after replacing color by kinematics

(N =4) × (N =4) → N =8 sugra(N =4) × (N =2) → N =6 sugra

(N =0) × (N =0) → Einstein gravity + axion+ dillaton

(N =4) × (N =0) → N =4 sugra

[BCJ]

Strings June 27 2011 H. Johansson 9

Four-point example

• Alternative decomposition, 4pt example

• Usual tree-level decomposition

gauge invariant

•Map

color structures

kinematic structures

color factors

kinematic numerators

absorbs 4-pt contact terms

Strings June 27 2011 H. Johansson 10

4-pt kinematic Jacobi relation

• Jacobi identity for color and for kinematics

color factors

• Kinematic numerators gauge dependent - but 4pt identity is gauge invariant

∼ gauge parameter

⇔

• Easy to check using Feynman rules

Strings June 27 2011 H. Johansson 11

Generalized gauge transformation…

Define “generalized gauge transformation” on amplitude as

To see the duality one must find the transformation that makes thenumerators obey the algebra -- in general a nontrivial task

Amplitudes invariant under this transformation, but not duality

such that

⁄ ⇔/

(2n-5)!! cubic diagrams [BCJ]

…explains why this kinematic structure has remained hidden.

Strings June 27 2011 H. Johansson 12

Tree-Level Evidence

Strings June 27 2011 H. Johansson 13

Duality gives new amplitude relations

4 points:

5 points:

…relations obtained for any multiplicity

sij..= (ki+kj+…)2

In color ordered tree amplitudes 3 legs can be fixed: (n-3)! basis

Bjerrum-Bohr, Damgaard, Vanhove; Stieberger

These were later found to be equivalent to monodromy relations on the open string worldsheet

Also field theory proofs through BCFW: Feng, Huang, Jia; Chen, Du, Feng

Strings June 27 2011 H. Johansson 14Right sector ⇔ modes in spacetime

Left sector ⇔ modes in spacetime

Tree-level gravity checks• Original conjecture checked through 8 points

⇔

⇔

Work by Tye and Zhang connects to heterotic string

Bern, Carrasco, HJ

double copyof YM

• All-multiplicity proof assuming gauge theory duality: Bern, Dennen, Huang, Kiermaier

Strings June 27 2011 H. Johansson 15

Some tree-level solutions

1 n

σ2 σn-1

•All-multiplicity solution for non-local tree numerators using KLT Kiermaier; Bjerrum-Bohr, Damgaard, Sondergaard Vanhove

•Explicit local numerators using pure-spinor methods in D=10 Mafra, Schlotterer, Stieberger

•Cubic Feynman rules obeying the duality for MHV sector of YM Monteiro and O'Connell

Strings June 27 2011 H. Johansson 16

Loop-Level Evidence

Strings June 27 2011 H. Johansson 17

Manifest duality in N = 4 SYM 4-pt ampl.

Duality: N = 8 SG is obtained if 1 → 2 (numerator squaring)

+ +4

31

22

41

3 3

21

4

K 1

s 1 s 1K 1

K

1-loop:

2-loop:

prefactor containshelicity structure:

Green, Schwarz,Brink (1982)

Bern, Dixon,Dunbar, Perelsteinand Rozowsky (1998)

Known cases of duality-satisfying loop amplitudes:

Strings June 27 2011 H. Johansson 18

Old form of 3-loop amplitudeProblem: no double copy in 0808.4112 [hep-th] (Bern, Carrasco, Dixon, HJ, Roiban)

N=4 SYM N=8 SG

Strings June 27 2011 H. Johansson 19

After nontrivial reshuffling3-loop N =4 SYM admits manifest realization of duality-- and N =8 SG is simply the square

1004.0476 [hep-th] Bern, Carrasco, HJ

Strings June 27 2011 H. Johansson 20

Works for non-susy theories

All-plus-helicity QCD amplitude:

+

+

+

+

= nDB + nBT + …

All-plus-helicity Einstein gravity amplitude:

++

++

++

++

= n2DB + n2

BT + …

(with dilation and axion in loops)

1004.0476 [hep-th] Bern, Carrasco, HJ

Strings June 27 2011 H. Johansson 21

1-loop 5-pts N =4 SYM and N =8 SG

e.g. Jacobi relation:

Carrasco, HJ 1106.4711 [hep-th]

-=

2

1

=

=

•The five-point amplitude makes the duality manifest ! •N =8 SG is obtained through the numerator double copy

Strings June 27 2011 H. Johansson 22

2-loop 5-pts N =4 SYM and N =8 SG

The 2-loop 5-pointamplitude withduality exposed

N = 8 SG obtainedfrom numeratordouble copies

Carrasco, HJ

1106.4711 [hep-th]

Strings June 27 2011 H. Johansson 23

Duality and UV behavior

Strings June 27 2011 H. Johansson 24

UV properties of N =8 supergravity

•Conventional superspace power counting forbids L=1,2divergences Green, Schwarz, Brink (1982), Howe and Stelle (1989), Marcus and Sagnotti (1985)

•Three-loop divergence ruled out by calculation: Bern, Carrasco, Dixon,HJ, Kosower, Roiban, (2007), Bern, Carrasco, Dixon, HJ, Roiban (2008)

•L<7 loop divergences ruled out by counterterm analysis, using E7(7)

symmetry and other methods, but a L=7 divergence is still possibleBeisert, Elvang, Freedman, Kiermaier, Morales, Stieberger; Björnsson, Green, Bossard, Howe,Stelle, Vanhove, Kallosh, Ramond

Quick status:

Through four loops the theories diverge in exactly the same dimension: Comparing N =8 SG and N =4 SYM UV behavior for D > 4

Bern, Carrasco, Dixon, HJ, Kosower, Roiban

Confirmed using duality-satisfying amplitudes: UV behavior is manifest Bern, Carrasco, Dixon, HJ, Roiban

Strings June 27 2011 H. Johansson 25

4-loops N =4 SYM and N =8 SGBern, Carrasco, Dixon, HJ, Roiban (to be published)

•85 diagrams in total•Duality manifest•Power counting manifest both N =4 and N =8•Both diverge in D=11/2

Strings June 27 2011 H. Johansson 26

Towards a kinematic Lie algebra

• In 1103.0312 [hep-th] Bern and Dennen investigate the tracestructure of kinematical numerators.

• In 1105.2565 [hep-th] Monteiro and O'Connell identify adiffeomorphism Lie algebra in the self-dual Yang-Mills sector.From this they obtain the kinematic structure constants forMHV tree amplitudes.

• In 1004.0693 [hep-th] Bern, Dennen, Huang, Kiermaier work outthe first terms of a duality-satisfying Lagrangian

All available evidence suggest that exist kinematic numerators ofgauge theory amplitudes that satisfy the same general algebraas the color structures of these theories.

Suggest the existence of a Lie algebra for the kinematics !

Strings June 27 2011 H. Johansson 27

Lagrangian formulation

• First attempt at Lagrangian with manifest duality

YM Lagrangian receives corrections at 5 points and higher

corrections proportional to the Jacobi identity (thus equal to zero)

1004.0693 [hep-th] Bern, Dennen, Huang, Kiermaier

Introduction of auxiliary fields gives local cubic Lagrangian

+ …

“squaring” gives gravity Lagrangian.

Strings June 27 2011 H. Johansson 28

SummaryGravity appears to be a double copy of Yang-Mills theory, order byorder in the S-matrix.

The double-copy structure in field theory becomes clear if kinematicnumerators are treated on equal footing with color factors. Suggestingthat a kinematic Lie algebra should exist.

Nontrivial evidence at tree and loop level supports the duality.

Lagrangian formulation, connection to string theory, give hints of futurepotential. Duality should be a key tool for nonplanar gauge theoryand gravity calculations.

What is the physical interpretation the duality ? What is the kinematic Liealgebra ? Further understanding of the connection to string theory mayhelp answer these questions.

Strings June 27 2011 H. Johansson 29

Extra slides

Strings June 27 2011 H. Johansson 30

Summary of checks of duality

(to be published: Bern, Carrasco, Dixon, HJ, Roiban)

Much evidence from: amplitude relations, KLT, string theory…

loops

legs

4

5678

0 1 2 3 4

⇑

N =4 SYM & N =8 SG

Less-SUSY theories: Tree level: all pure gauge theories have the same

tree amplitudes as N =4 SYM

Two-loop N =0 YM, 4p all-plus helicity

(to be published: Carrasco, HJ)

Strings June 27 2011 H. Johansson 31

3-loop 5-pts N =4 SYM and N =8 SG

(“ladder-like” diagrams)

Carrasco, HJ(to be published)

Again the D-dimensional amplitude admits arepresentation with manifest duality

N = 8 SG obtained fromsquaring the numerators

Strings June 27 2011 H. Johansson 32

3-loop 5-pts N =4 SYM and N =8 SGsome “Mercedes-like” diagrams… Carrasco, HJ

(to be published)

Strings June 27 2011 H. Johansson 33

3-loop 5-pts N =4 SYM and N =8 SGCarrasco, HJ(to be published)

…in total 42 diagrams.

Conveniently the UV divergent diagrams (in D=6) are very simple:

(for SG the UV div. comes from the other diagrams as well)

Strings June 27 2011 H. Johansson 34

Unitarity

Optical theorem:

The unitarity method reconstructs the amplitudes avoiding dispersion relationsBern, Dixon, Dunbar, Kosower (1994)

+ + + =

Compute a cut: put loop legs on-shell in amplitude = sew trees amplitudes

checking every cut channel will fix the loop integrals

Strings June 27 2011 H. Johansson 35

Amplitude relations for any number of legs

•General relations for gauge theory partial amplitudes

where

and

and

and tk is the position of leg k in the set {ρ}

Αn(σ1,σ2,..,σn) = α1 Αn(1,2,..,n) + α2 Αn(2,1,..,n) + … + α(n-3)! Αn(3,2,..,n)

Basis size: (n-3)! Compare to Kleiss-Kuijf relations (n-2)!

Bern, Carrasco, HJ

Recent proofs: Bjerrum-Bohr, Damgaard, Vanhove; Feng, Huang, Jia

Strings June 27 2011 H. Johansson 36

Gauge theory amplitude properties

• Well-known partial amplitude properties

• Tree level, adjoint representation

cyclic symmetry

• New relations reduce independent basis to (n - 3)!

(n - 2)!

reflection symmetry

Kleiss-Kuijfrelations

(n -1)!/2

“photon”-decoupling identity

gauge invariant

Bern, Carrasco, HJ